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The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation. (English) Zbl 0674.35027
Existence of positive weak solutions $$u\in C^{\infty}(S^ n\setminus \Omega)\cap L^ p(S^ n)$$ of $Lu+(n(n-2)/4)u^{(n+2)/(n-2)}=0\quad on\quad S^ n={\mathbb{R}}^ n\cup \{\infty \}\quad (n\geq 3),$ is proved, having singularities on a compact set $$\Omega \subset {\mathbb{R}}^ n$$. Here the linear operator L is defined by $$Lu=\Delta_{g_ 0}-(n(n-2)/4)u$$ where $$g_ 0$$ denotes the standard Riemannian metric on $$S^ n$$. The existence theorem holds for a remarkable variety of totally disconnected sets $$\Omega$$ of small Hausdorff dimension, $$\Omega$$ may be e.g. any finite collection of at least two points.
Reviewer: L.Simon

##### MSC:
 35J60 Nonlinear elliptic equations 53B20 Local Riemannian geometry 35A20 Analyticity in context of PDEs 35D05 Existence of generalized solutions of PDE (MSC2000) 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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##### References:
 [1] Aviles, Comm. Math. Phys. 108 pp 177– (1987) [2] Donaldson, J. Diff. Geom. 24 pp 275– (1986) [3] to appear in ICM86 proceedings. [4] Gidas, Comm. Math. Phys. 68 pp 209– (1979) [5] Gidas, Comm. Pure Applied Math. 34 pp 525– (1981) [6] and , Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, Heidelberg, New York, 1977. [7] Kazdan, Ann. of Math. 99 pp 14– (1974) [8] Topics in Nonlinear Functional Analysis, NYU Lecture Notes, 1973,. [9] Pohozaev, Soviet Math. Doklady 6 pp 1408– (1965) [10] and , Functional Analysis, Ungar, New York, 1955. [11] Schoen, J. Diff. Geom. 20 pp 479– (1984) [12] and , Conformally flat manifolds, Kleinian groups, and scalar curvature, to appear in Invent. Math., 1988,. [13] Serrin, Arch. Rat. Mech. Anal. 20 pp 163– (1965) [14] Taubes, J. Diff. Geom. 19 pp 517– (1984)
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